A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk

Transcription

1 A Few Goo Distance-Rate-Time Problems #6 of Gottschalk s Gestalts A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk Infinite Vistas Press PVD RI 00 GG6- (4)

2 00 Walter Gottschalk 500 Angell St #44 Provience RI 0906 permission is grante without charge to reprouce & istribute this item at cost for eucational purposes; attribution requeste; no warranty of infallibility is posite GG6-

3 consier a moving particle, mathematically represente by a point, in uniform (= constant) spee along a straight line (= in rectilinear motion) or along a smooth curve (= in curvilinear motion) for a specifie finite amount of time istance = istance the particle has move along its orbit / path = istance n rate = the constant rate of spee of the particle along its orbit / path = r rate n time = the time of uration of the particle's motion along its orbit / path = t t ime GG6-3 n

4 it is clear from physical consierations that: istance equals rate times time = r t = rt rate equals istance ivie by time r = t r = t time equals istance ivie by rate t = r t = r GG6-4

5 given any two of the quantities istance = rate = r time = t the thir is uniquely etermine by one of the three above formulas below is a list of five istance - rate - time problems that are nonroutine challenges of varing ifficulty GG6-5

6 P. Rowing Upsteam & Downstream A person can row a boat six miles per hour in still water. In a river where the current is two miles per hour, it takes him thirty minutes longer to row a certain istance upstream than to row the same istance ownstream. Fin how long it takes him to row upstream, how long to row ownstream, an how many miles he rows. S. rowing upstream & ownstream let enote the istance rowe in one irection he rows 6 - = 4 mph upstream & he rows 6 + = 8 mph ownstream hence the time rowing upstream is 4 hours & the time rowing ownstream is 8 hours GG6-6

7 then = & = 4 miles time to row upstream is one hour; time to row ownstream is thirty minues; total istance rowe is eight miles GG6-7

8 P. The Rowboat & the Beach Ball A person in a rowboat on a calmly flowing river rops a beach ball in the river an then rows upstream for ten minutes. He immeiately turns aroun an rows ownstream to overtake an pick up the ball. He then notices that the ball has floate ownstream for exactly one mile from the rop-off point. How fast is the river flowing? S. the rowboat & the beach ball thinking of relative motion on the river itself, the oarsman rops the ball on a stationary stream an rows away for 0 minutes ; he then rows back to the ball an this last trip must also take 0 minutes in that 0 minute perio the river has move one mile; hence the current of the river is 3 miles per hour GG6-8

9 P3. Two Trains & a Fly Two trains are heaing towar one another on parallel tracks an are originally a istance apart of 00 miles. One train is traveling 40 miles per hour. The other train is traveling 60 miles per hour. A fast inefatigible fly starting from one train at the beginning flies back an forth at a spee of 80 miles per hour from one train to the other train continuously, first touching one train an then the other without rest until the trains meet. How far i the fly fly? S3. two trains & a fly the trains meet in hours; the fly then flies for hours the fly files at a spee of 80 mph hence the fly flies times 80 = 60 miles GG6-9

10 P4. The Passenger Train & the Freight Train A passenger train is x times faster than a freight train. The passenger train takes x times as long to overtake the freight train when going in the same irection as it takes the two trains to pass when going in opposite irections. Fin x. S4. the passenger train & the freight train assume the spee of the freight train to be unity & assume the sum of the lengths of the trains to be unity; this then etermines the unit of time; this simpifies the algebraic calculation their relative spee when going in the same irection is x -; their relative spee when going in opposite irections is x + ; the overtaking/passing istance in the same/opposite irection(s) is ; GG6-0

11 the passenger train takes x - to overtake in the same irection; the pasenger train takes x + to pass in the opposite irection time units time units hence x = x - x + & x = + = GG6-

12 P5. The Army & the Courier An army ten miles long is marching along a wining mountain roa. A motorcycle courier elivers a message from the rear of the army to the front of the army an immeiatley returns to the rear. He notices that when he returns to the rear that he is at the point where the front of the army was when he starte out. How far i the courier travel? S5. the army & the courier for two uniform motions uring two time intervals t an t, = r t = r t ' = r t' ' = r t' whence GG6-

13 & ' ' = r r = r r = ' ' let the length of the army be the unit of length to simplify the little algebraic manipulations let x be the istance the courier travele forwar from the position of the army front at outset to the position of the army front at elivery; the total istance covere by the courier is x + GG6-3

14 using the two time intervals uring which the courier travele from original back to elivery front & from elivery front to final back (= original front position) an also using the proportion establishe above + x = x whence x & = x + x -x = + = hence the total istance the courier travele is ( ) = miles GG6-4